How to curve the edges of a plane?
Note: This is non destructive way to do such thing. You can always use
  Proportional Editing to directly model this shape.



Add a Plane and Subdivide it as much as you need/want.





Add Lattice and Scale it by 2.
Go to Lattice Data tab and set U: 3, V: 3, W: 1.





Add Lattice modifier to the Plane and choose Lattice as Object.





Go back to Lattice in Edit Mode, select needed vertices and Scale them down and also move center vertex on the Z axis.




Blend file: